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\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\title{复变函数练习3.1 - 复积分的概念 }
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\date{2024 年 4 月 8 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}

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\item  %Problem 01
设 $C: z=z(t), \alpha\le t\le\beta$ 是复平面中的一条逐段光滑的有向曲线，设复值函数 $f(z)$ 沿着 $C$ 有定义，
写出复积分的定义， $$\int_C f(z)dz. $$

\vspace{0.0cm}

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\item  %Problem 02
设函数 $f(z)=u(x,y)+iv(x,y)$ 沿曲线 $C$ 连续，证明 $f(z)$ 沿 $C$ 可积，且 
$$\int_C f(z)dz = \int_C udx-vdy + i\int_C vdx+udy. $$

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\item  %Problem 03
设 $C$ 是连接复平面上的点 $a$ 和点 $b$ 的任一曲线，计算下述复积分，
$$(1) \int_Cdz,\hspace{0.5cm} (2) \int_Czdz.$$

 
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\item  %Problem 04
设 $n$ 是一个整数，设 $C$ 是以 $a$ 为圆心，以 $\rho$ 为半径的圆周，计算积分
$$ \int_C \frac{dz}{(z-a)^n}.$$

\vspace{0.0cm}

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\item  %Problem 05
证明复积分的基本不等式 $$\left\lvert \int_C f(z)dz\right \rvert \le \int_C \left\lvert f(z) \right\lvert \left\lvert dz \right\lvert. $$

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\item  %Problem 06
设 $C$ 是连接点 $i$ 和点 $2+i$ 的直线段，证明 $$\left\lvert \int_C \frac{dz}{z^2}\right \rvert \le 2. $$
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\item  %Problem 07
设 $r>0$ 以及 $|a|\neq r$, 设 $C$ 是圆周 $|z|=r$, 证明不等式 $$\left\lvert \int_C \frac{dz}{(z-a)(z+a)} \right \rvert 
< \frac{ 2\pi r }{ \left\lvert r^2-|a|^2 \right\lvert}. $$

\vspace{0.0cm}

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\item  %Problem 08
设积分路径 $C$ 分别如下，
\begin{enumerate}
\item  连接点 0 到点 $1+i$ 的直线段。
\item  连接点 0 到点 1 的直线段，以及连接点 1 到点 $1+i$ 的直线段。 
\end{enumerate}
计算积分 $$\int_C \mathrm{Re}(z)dz. $$

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\item  %Problem 09
设积分路径 $C$ 是连接点 0 和点 $1+i$ 的直线段，计算积分 $$\int_C (x-y+ix^2)dz. $$
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\item  %Problem 10
设积分路径 $C$ 分别是 (1)直线段、(2)上半单位圆周、(3)下半单位圆周，计算积分 $$\int_{-1}^{1} zdz. $$
\vspace{0.0cm}


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\end{enumerate}


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\end{document}

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